Theorem fsuppco2 8463

Description: The composition of a function which maps the zero to zero with a finitely supported function is finitely supported. This is not only a special case of fsuppcor 8464 because it does not require that the "zero" is an element of the range of the finitely supported function. (Contributed by AV, 6-Jun-2019.)

Hypotheses

Ref	Expression
fsuppco2.z	⊢ (𝜑 → 𝑍 ∈ 𝑊)
fsuppco2.f	⊢ (𝜑 → 𝐹:𝐴⟶𝐵)
fsuppco2.g	⊢ (𝜑 → 𝐺:𝐵⟶𝐵)
fsuppco2.a	⊢ (𝜑 → 𝐴 ∈ 𝑈)
fsuppco2.b	⊢ (𝜑 → 𝐵 ∈ 𝑉)
fsuppco2.n	⊢ (𝜑 → 𝐹 finSupp 𝑍)
fsuppco2.i	⊢ (𝜑 → (𝐺‘𝑍) = 𝑍)

Assertion

Ref	Expression
fsuppco2	⊢ (𝜑 → (𝐺 ∘ 𝐹) finSupp 𝑍)

Proof of Theorem fsuppco2

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		fsuppco2.g	. . . 4 ⊢ (𝜑 → 𝐺:𝐵⟶𝐵)
2		ffun 6188	. . . 4 ⊢ (𝐺:𝐵⟶𝐵 → Fun 𝐺)
3	1, 2	syl 17	. . 3 ⊢ (𝜑 → Fun 𝐺)
4		fsuppco2.f	. . . 4 ⊢ (𝜑 → 𝐹:𝐴⟶𝐵)
5		ffun 6188	. . . 4 ⊢ (𝐹:𝐴⟶𝐵 → Fun 𝐹)
6	4, 5	syl 17	. . 3 ⊢ (𝜑 → Fun 𝐹)
7		funco 6071	. . 3 ⊢ ((Fun 𝐺 ∧ Fun 𝐹) → Fun (𝐺 ∘ 𝐹))
8	3, 6, 7	syl2anc 565	. 2 ⊢ (𝜑 → Fun (𝐺 ∘ 𝐹))
9		fsuppco2.n	. . . 4 ⊢ (𝜑 → 𝐹 finSupp 𝑍)
10	9	fsuppimpd 8437	. . 3 ⊢ (𝜑 → (𝐹 supp 𝑍) ∈ Fin)
11		fco 6198	. . . . 5 ⊢ ((𝐺:𝐵⟶𝐵 ∧ 𝐹:𝐴⟶𝐵) → (𝐺 ∘ 𝐹):𝐴⟶𝐵)
12	1, 4, 11	syl2anc 565	. . . 4 ⊢ (𝜑 → (𝐺 ∘ 𝐹):𝐴⟶𝐵)
13		eldifi 3881	. . . . . 6 ⊢ (𝑥 ∈ (𝐴 ∖ (𝐹 supp 𝑍)) → 𝑥 ∈ 𝐴)
14		fvco3 6417	. . . . . 6 ⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝑥 ∈ 𝐴) → ((𝐺 ∘ 𝐹)‘𝑥) = (𝐺‘(𝐹‘𝑥)))
15	4, 13, 14	syl2an 575	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴 ∖ (𝐹 supp 𝑍))) → ((𝐺 ∘ 𝐹)‘𝑥) = (𝐺‘(𝐹‘𝑥)))
16		ssid 3771	. . . . . . . 8 ⊢ (𝐹 supp 𝑍) ⊆ (𝐹 supp 𝑍)
17	16	a1i 11	. . . . . . 7 ⊢ (𝜑 → (𝐹 supp 𝑍) ⊆ (𝐹 supp 𝑍))
18		fsuppco2.a	. . . . . . 7 ⊢ (𝜑 → 𝐴 ∈ 𝑈)
19		fsuppco2.z	. . . . . . 7 ⊢ (𝜑 → 𝑍 ∈ 𝑊)
20	4, 17, 18, 19	suppssr 7477	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴 ∖ (𝐹 supp 𝑍))) → (𝐹‘𝑥) = 𝑍)
21	20	fveq2d 6336	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴 ∖ (𝐹 supp 𝑍))) → (𝐺‘(𝐹‘𝑥)) = (𝐺‘𝑍))
22		fsuppco2.i	. . . . . 6 ⊢ (𝜑 → (𝐺‘𝑍) = 𝑍)
23	22	adantr 466	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴 ∖ (𝐹 supp 𝑍))) → (𝐺‘𝑍) = 𝑍)
24	15, 21, 23	3eqtrd 2808	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴 ∖ (𝐹 supp 𝑍))) → ((𝐺 ∘ 𝐹)‘𝑥) = 𝑍)
25	12, 24	suppss 7476	. . 3 ⊢ (𝜑 → ((𝐺 ∘ 𝐹) supp 𝑍) ⊆ (𝐹 supp 𝑍))
26		ssfi 8335	. . 3 ⊢ (((𝐹 supp 𝑍) ∈ Fin ∧ ((𝐺 ∘ 𝐹) supp 𝑍) ⊆ (𝐹 supp 𝑍)) → ((𝐺 ∘ 𝐹) supp 𝑍) ∈ Fin)
27	10, 25, 26	syl2anc 565	. 2 ⊢ (𝜑 → ((𝐺 ∘ 𝐹) supp 𝑍) ∈ Fin)
28		fsuppco2.b	. . . . 5 ⊢ (𝜑 → 𝐵 ∈ 𝑉)
29		fex 6632	. . . . 5 ⊢ ((𝐺:𝐵⟶𝐵 ∧ 𝐵 ∈ 𝑉) → 𝐺 ∈ V)
30	1, 28, 29	syl2anc 565	. . . 4 ⊢ (𝜑 → 𝐺 ∈ V)
31		fex 6632	. . . . 5 ⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝐴 ∈ 𝑈) → 𝐹 ∈ V)
32	4, 18, 31	syl2anc 565	. . . 4 ⊢ (𝜑 → 𝐹 ∈ V)
33		coexg 7263	. . . 4 ⊢ ((𝐺 ∈ V ∧ 𝐹 ∈ V) → (𝐺 ∘ 𝐹) ∈ V)
34	30, 32, 33	syl2anc 565	. . 3 ⊢ (𝜑 → (𝐺 ∘ 𝐹) ∈ V)
35		isfsupp 8434	. . 3 ⊢ (((𝐺 ∘ 𝐹) ∈ V ∧ 𝑍 ∈ 𝑊) → ((𝐺 ∘ 𝐹) finSupp 𝑍 ↔ (Fun (𝐺 ∘ 𝐹) ∧ ((𝐺 ∘ 𝐹) supp 𝑍) ∈ Fin)))
36	34, 19, 35	syl2anc 565	. 2 ⊢ (𝜑 → ((𝐺 ∘ 𝐹) finSupp 𝑍 ↔ (Fun (𝐺 ∘ 𝐹) ∧ ((𝐺 ∘ 𝐹) supp 𝑍) ∈ Fin)))
37	8, 27, 36	mpbir2and 684	1 ⊢ (𝜑 → (𝐺 ∘ 𝐹) finSupp 𝑍)

Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 382   = wceq 1630   ∈ wcel 2144  Vcvv 3349   ∖ cdif 3718   ⊆ wss 3721   class class class wbr 4784   ∘ ccom 5253  Fun wfun 6025  ⟶wf 6027  ‘cfv 6031  (class class class)co 6792   supp csupp 7445  Fincfn 8108   finSupp cfsupp 8430

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1990  ax-6 2056  ax-7 2092  ax-8 2146  ax-9 2153  ax-10 2173  ax-11 2189  ax-12 2202  ax-13 2407  ax-ext 2750  ax-rep 4902  ax-sep 4912  ax-nul 4920  ax-pow 4971  ax-pr 5034  ax-un 7095

This theorem depends on definitions:  df-bi 197  df-an 383  df-or 827  df-3or 1071  df-3an 1072  df-tru 1633  df-ex 1852  df-nf 1857  df-sb 2049  df-eu 2621  df-mo 2622  df-clab 2757  df-cleq 2763  df-clel 2766  df-nfc 2901  df-ne 2943  df-ral 3065  df-rex 3066  df-reu 3067  df-rab 3069  df-v 3351  df-sbc 3586  df-csb 3681  df-dif 3724  df-un 3726  df-in 3728  df-ss 3735  df-pss 3737  df-nul 4062  df-if 4224  df-pw 4297  df-sn 4315  df-pr 4317  df-tp 4319  df-op 4321  df-uni 4573  df-iun 4654  df-br 4785  df-opab 4845  df-mpt 4862  df-tr 4885  df-id 5157  df-eprel 5162  df-po 5170  df-so 5171  df-fr 5208  df-we 5210  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-ord 5869  df-on 5870  df-lim 5871  df-suc 5872  df-iota 5994  df-fun 6033  df-fn 6034  df-f 6035  df-f1 6036  df-fo 6037  df-f1o 6038  df-fv 6039  df-ov 6795  df-oprab 6796  df-mpt2 6797  df-om 7212  df-supp 7446  df-er 7895  df-en 8109  df-fin 8112  df-fsupp 8431

This theorem is referenced by:  gsumzinv  18551  gsumsub  18554